Properties

Label 560.2.e.c.559.7
Level $560$
Weight $2$
Character 560.559
Analytic conductor $4.472$
Analytic rank $0$
Dimension $8$
CM discriminant -35
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [560,2,Mod(559,560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("560.559");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 560.e (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.47162251319\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.121550625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 559.7
Root \(-0.862555 - 0.141174i\) of defining polynomial
Character \(\chi\) \(=\) 560.559
Dual form 560.2.e.c.559.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.25937i q^{3} -2.23607 q^{5} +2.64575i q^{7} -7.62348 q^{9} +O(q^{10})\) \(q+3.25937i q^{3} -2.23607 q^{5} +2.64575i q^{7} -7.62348 q^{9} -0.359964i q^{11} +5.64539 q^{13} -7.28817i q^{15} -7.99190 q^{17} -8.62348 q^{21} +5.00000 q^{25} -15.0696i q^{27} -0.623475 q^{29} +1.17325 q^{33} -5.91608i q^{35} +18.4004i q^{39} +17.0466 q^{45} +5.71383i q^{47} -7.00000 q^{49} -26.0485i q^{51} +0.804903i q^{55} -20.1698i q^{63} -12.6235 q^{65} +11.8322i q^{71} -13.4164 q^{73} +16.2968i q^{75} +0.952374 q^{77} +8.00809i q^{79} +26.2470 q^{81} +15.8745i q^{83} +17.8704 q^{85} -2.03214i q^{87} +14.9363i q^{91} -12.6849 q^{97} +2.74417i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 20 q^{9} - 28 q^{21} + 40 q^{25} + 36 q^{29} - 56 q^{49} - 60 q^{65} + 128 q^{81} + 20 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/560\mathbb{Z}\right)^\times\).

\(n\) \(241\) \(337\) \(351\) \(421\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.25937i 1.88180i 0.338689 + 0.940898i \(0.390016\pi\)
−0.338689 + 0.940898i \(0.609984\pi\)
\(4\) 0 0
\(5\) −2.23607 −1.00000
\(6\) 0 0
\(7\) 2.64575i 1.00000i
\(8\) 0 0
\(9\) −7.62348 −2.54116
\(10\) 0 0
\(11\) − 0.359964i − 0.108533i −0.998526 0.0542666i \(-0.982718\pi\)
0.998526 0.0542666i \(-0.0172821\pi\)
\(12\) 0 0
\(13\) 5.64539 1.56575 0.782875 0.622179i \(-0.213753\pi\)
0.782875 + 0.622179i \(0.213753\pi\)
\(14\) 0 0
\(15\) − 7.28817i − 1.88180i
\(16\) 0 0
\(17\) −7.99190 −1.93832 −0.969160 0.246433i \(-0.920742\pi\)
−0.969160 + 0.246433i \(0.920742\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −8.62348 −1.88180
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) − 15.0696i − 2.90015i
\(28\) 0 0
\(29\) −0.623475 −0.115776 −0.0578882 0.998323i \(-0.518437\pi\)
−0.0578882 + 0.998323i \(0.518437\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 1.17325 0.204237
\(34\) 0 0
\(35\) − 5.91608i − 1.00000i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 18.4004i 2.94642i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 17.0466 2.54116
\(46\) 0 0
\(47\) 5.71383i 0.833448i 0.909033 + 0.416724i \(0.136822\pi\)
−0.909033 + 0.416724i \(0.863178\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) − 26.0485i − 3.64752i
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0.804903i 0.108533i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) − 20.1698i − 2.54116i
\(64\) 0 0
\(65\) −12.6235 −1.56575
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 11.8322i 1.40422i 0.712069 + 0.702109i \(0.247758\pi\)
−0.712069 + 0.702109i \(0.752242\pi\)
\(72\) 0 0
\(73\) −13.4164 −1.57027 −0.785136 0.619324i \(-0.787407\pi\)
−0.785136 + 0.619324i \(0.787407\pi\)
\(74\) 0 0
\(75\) 16.2968i 1.88180i
\(76\) 0 0
\(77\) 0.952374 0.108533
\(78\) 0 0
\(79\) 8.00809i 0.900981i 0.892781 + 0.450490i \(0.148751\pi\)
−0.892781 + 0.450490i \(0.851249\pi\)
\(80\) 0 0
\(81\) 26.2470 2.91633
\(82\) 0 0
\(83\) 15.8745i 1.74245i 0.490881 + 0.871227i \(0.336675\pi\)
−0.490881 + 0.871227i \(0.663325\pi\)
\(84\) 0 0
\(85\) 17.8704 1.93832
\(86\) 0 0
\(87\) − 2.03214i − 0.217868i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 14.9363i 1.56575i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −12.6849 −1.28796 −0.643979 0.765043i \(-0.722718\pi\)
−0.643979 + 0.765043i \(0.722718\pi\)
\(98\) 0 0
\(99\) 2.74417i 0.275800i
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 18.7513i 1.84762i 0.382851 + 0.923810i \(0.374942\pi\)
−0.382851 + 0.923810i \(0.625058\pi\)
\(104\) 0 0
\(105\) 19.2827 1.88180
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 9.87043 0.945415 0.472708 0.881219i \(-0.343277\pi\)
0.472708 + 0.881219i \(0.343277\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −43.0375 −3.97882
\(118\) 0 0
\(119\) − 21.1446i − 1.93832i
\(120\) 0 0
\(121\) 10.8704 0.988221
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.1803 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 33.6967i 2.90015i
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) −18.6235 −1.56838
\(142\) 0 0
\(143\) − 2.03214i − 0.169936i
\(144\) 0 0
\(145\) 1.39413 0.115776
\(146\) 0 0
\(147\) − 22.8156i − 1.88180i
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) − 5.84831i − 0.475929i −0.971274 0.237964i \(-0.923520\pi\)
0.971274 0.237964i \(-0.0764802\pi\)
\(152\) 0 0
\(153\) 60.9260 4.92558
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 13.4164 1.07075 0.535373 0.844616i \(-0.320171\pi\)
0.535373 + 0.844616i \(0.320171\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) −2.62348 −0.204237
\(166\) 0 0
\(167\) − 25.2700i − 1.95545i −0.209881 0.977727i \(-0.567308\pi\)
0.209881 0.977727i \(-0.432692\pi\)
\(168\) 0 0
\(169\) 18.8704 1.45157
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 15.0314 1.14282 0.571409 0.820666i \(-0.306397\pi\)
0.571409 + 0.820666i \(0.306397\pi\)
\(174\) 0 0
\(175\) 13.2288i 1.00000i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 11.8322i − 0.884377i −0.896922 0.442189i \(-0.854202\pi\)
0.896922 0.442189i \(-0.145798\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.87679i 0.210372i
\(188\) 0 0
\(189\) 39.8704 2.90015
\(190\) 0 0
\(191\) 20.4246i 1.47788i 0.673774 + 0.738938i \(0.264672\pi\)
−0.673774 + 0.738938i \(0.735328\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) − 41.1445i − 2.94642i
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 1.64956i − 0.115776i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 28.7927i 1.98217i 0.133226 + 0.991086i \(0.457467\pi\)
−0.133226 + 0.991086i \(0.542533\pi\)
\(212\) 0 0
\(213\) −38.5654 −2.64245
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) − 43.7290i − 2.95493i
\(220\) 0 0
\(221\) −45.1174 −3.03492
\(222\) 0 0
\(223\) − 20.3611i − 1.36348i −0.731594 0.681740i \(-0.761223\pi\)
0.731594 0.681740i \(-0.238777\pi\)
\(224\) 0 0
\(225\) −38.1174 −2.54116
\(226\) 0 0
\(227\) 21.2058i 1.40748i 0.710460 + 0.703738i \(0.248487\pi\)
−0.710460 + 0.703738i \(0.751513\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 3.10414i 0.204237i
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) − 12.7765i − 0.833448i
\(236\) 0 0
\(237\) −26.1013 −1.69546
\(238\) 0 0
\(239\) − 22.5844i − 1.46087i −0.682985 0.730433i \(-0.739318\pi\)
0.682985 0.730433i \(-0.260682\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 40.3396i 2.58779i
\(244\) 0 0
\(245\) 15.6525 1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −51.7409 −3.27894
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 58.2463i 3.64752i
\(256\) 0 0
\(257\) 4.47214 0.278964 0.139482 0.990225i \(-0.455456\pi\)
0.139482 + 0.990225i \(0.455456\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 4.75305 0.294206
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) −48.6829 −2.94642
\(274\) 0 0
\(275\) − 1.79982i − 0.108533i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 11.3765 0.678667 0.339333 0.940666i \(-0.389799\pi\)
0.339333 + 0.940666i \(0.389799\pi\)
\(282\) 0 0
\(283\) − 27.7245i − 1.64805i −0.566553 0.824025i \(-0.691723\pi\)
0.566553 0.824025i \(-0.308277\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 46.8704 2.75708
\(290\) 0 0
\(291\) − 41.3448i − 2.42367i
\(292\) 0 0
\(293\) 32.9200 1.92320 0.961602 0.274446i \(-0.0884946\pi\)
0.961602 + 0.274446i \(0.0884946\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −5.42451 −0.314762
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 11.3879i 0.649942i 0.945724 + 0.324971i \(0.105355\pi\)
−0.945724 + 0.324971i \(0.894645\pi\)
\(308\) 0 0
\(309\) −61.1174 −3.47685
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −35.2665 −1.99338 −0.996689 0.0813030i \(-0.974092\pi\)
−0.996689 + 0.0813030i \(0.974092\pi\)
\(314\) 0 0
\(315\) 45.1011i 2.54116i
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0.224428i 0.0125656i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 28.2269 1.56575
\(326\) 0 0
\(327\) 32.1713i 1.77908i
\(328\) 0 0
\(329\) −15.1174 −0.833448
\(330\) 0 0
\(331\) 35.4965i 1.95106i 0.219860 + 0.975531i \(0.429440\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 18.5203i − 1.00000i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) − 85.0738i − 4.54090i
\(352\) 0 0
\(353\) 6.08715 0.323986 0.161993 0.986792i \(-0.448208\pi\)
0.161993 + 0.986792i \(0.448208\pi\)
\(354\) 0 0
\(355\) − 26.4575i − 1.40422i
\(356\) 0 0
\(357\) 68.9179 3.64752
\(358\) 0 0
\(359\) 11.8322i 0.624477i 0.950004 + 0.312239i \(0.101079\pi\)
−0.950004 + 0.312239i \(0.898921\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 35.4307i 1.85963i
\(364\) 0 0
\(365\) 30.0000 1.57027
\(366\) 0 0
\(367\) − 38.3075i − 1.99964i −0.0190919 0.999818i \(-0.506077\pi\)
0.0190919 0.999818i \(-0.493923\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) − 36.4408i − 1.88180i
\(376\) 0 0
\(377\) −3.51976 −0.181277
\(378\) 0 0
\(379\) 35.4965i 1.82333i 0.410932 + 0.911666i \(0.365203\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 15.8745i − 0.811149i −0.914062 0.405575i \(-0.867071\pi\)
0.914062 0.405575i \(-0.132929\pi\)
\(384\) 0 0
\(385\) −2.12957 −0.108533
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −24.6235 −1.24846 −0.624230 0.781241i \(-0.714587\pi\)
−0.624230 + 0.781241i \(0.714587\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 17.9066i − 0.900981i
\(396\) 0 0
\(397\) 19.7244 0.989941 0.494971 0.868910i \(-0.335179\pi\)
0.494971 + 0.868910i \(0.335179\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −39.1174 −1.95343 −0.976714 0.214544i \(-0.931173\pi\)
−0.976714 + 0.214544i \(0.931173\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −58.6900 −2.91633
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) − 35.4965i − 1.74245i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 33.8704 1.65074 0.825372 0.564590i \(-0.190966\pi\)
0.825372 + 0.564590i \(0.190966\pi\)
\(422\) 0 0
\(423\) − 43.5593i − 2.11792i
\(424\) 0 0
\(425\) −39.9595 −1.93832
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 6.62348 0.319784
\(430\) 0 0
\(431\) 23.3044i 1.12253i 0.827636 + 0.561266i \(0.189685\pi\)
−0.827636 + 0.561266i \(0.810315\pi\)
\(432\) 0 0
\(433\) 40.2492 1.93425 0.967127 0.254293i \(-0.0818429\pi\)
0.967127 + 0.254293i \(0.0818429\pi\)
\(434\) 0 0
\(435\) 4.54399i 0.217868i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 53.3643 2.54116
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 19.5562i 0.924977i
\(448\) 0 0
\(449\) 40.3643 1.90491 0.952455 0.304679i \(-0.0985491\pi\)
0.952455 + 0.304679i \(0.0985491\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 19.0618 0.895601
\(454\) 0 0
\(455\) − 33.3986i − 1.56575i
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 120.435i 5.62141i
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 40.7620i − 1.88624i −0.332454 0.943119i \(-0.607877\pi\)
0.332454 0.943119i \(-0.392123\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 43.7290i 2.01493i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 28.3643 1.28796
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 43.3690i − 1.95722i −0.205731 0.978609i \(-0.565957\pi\)
0.205731 0.978609i \(-0.434043\pi\)
\(492\) 0 0
\(493\) 4.98275 0.224412
\(494\) 0 0
\(495\) − 6.13616i − 0.275800i
\(496\) 0 0
\(497\) −31.3050 −1.40422
\(498\) 0 0
\(499\) − 26.6329i − 1.19225i −0.802890 0.596127i \(-0.796706\pi\)
0.802890 0.596127i \(-0.203294\pi\)
\(500\) 0 0
\(501\) 82.3643 3.67977
\(502\) 0 0
\(503\) 44.8262i 1.99870i 0.0360049 + 0.999352i \(0.488537\pi\)
−0.0360049 + 0.999352i \(0.511463\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 61.5057i 2.73156i
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) − 35.4965i − 1.57027i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 41.9292i − 1.84762i
\(516\) 0 0
\(517\) 2.05677 0.0904567
\(518\) 0 0
\(519\) 48.9929i 2.15055i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 37.0405i 1.61967i 0.586659 + 0.809834i \(0.300443\pi\)
−0.586659 + 0.809834i \(0.699557\pi\)
\(524\) 0 0
\(525\) −43.1174 −1.88180
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 38.5654 1.66422
\(538\) 0 0
\(539\) 2.51975i 0.108533i
\(540\) 0 0
\(541\) −6.12957 −0.263531 −0.131765 0.991281i \(-0.542065\pi\)
−0.131765 + 0.991281i \(0.542065\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −22.0709 −0.945415
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −21.1874 −0.900981
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −9.37652 −0.395877
\(562\) 0 0
\(563\) 15.8745i 0.669031i 0.942390 + 0.334515i \(0.108573\pi\)
−0.942390 + 0.334515i \(0.891427\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 69.4429i 2.91633i
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) 35.4965i 1.48548i 0.669579 + 0.742741i \(0.266474\pi\)
−0.669579 + 0.742741i \(0.733526\pi\)
\(572\) 0 0
\(573\) −66.5714 −2.78106
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 46.5573 1.93820 0.969102 0.246661i \(-0.0793334\pi\)
0.969102 + 0.246661i \(0.0793334\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −42.0000 −1.74245
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 96.2348 3.97882
\(586\) 0 0
\(587\) − 47.6235i − 1.96563i −0.184585 0.982817i \(-0.559094\pi\)
0.184585 0.982817i \(-0.440906\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 41.8642 1.71916 0.859579 0.511003i \(-0.170726\pi\)
0.859579 + 0.511003i \(0.170726\pi\)
\(594\) 0 0
\(595\) 47.2807i 1.93832i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 18.9848i 0.775698i 0.921723 + 0.387849i \(0.126782\pi\)
−0.921723 + 0.387849i \(0.873218\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −24.3070 −0.988221
\(606\) 0 0
\(607\) 39.9173i 1.62019i 0.586296 + 0.810097i \(0.300586\pi\)
−0.586296 + 0.810097i \(0.699414\pi\)
\(608\) 0 0
\(609\) 5.37652 0.217868
\(610\) 0 0
\(611\) 32.2568i 1.30497i
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 49.5773i 1.97364i 0.161817 + 0.986821i \(0.448265\pi\)
−0.161817 + 0.986821i \(0.551735\pi\)
\(632\) 0 0
\(633\) −93.8460 −3.73004
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −39.5177 −1.56575
\(638\) 0 0
\(639\) − 90.2022i − 3.56834i
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) 0 0
\(643\) − 30.9441i − 1.22032i −0.792279 0.610158i \(-0.791106\pi\)
0.792279 0.610158i \(-0.208894\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 47.6235i 1.87227i 0.351636 + 0.936137i \(0.385626\pi\)
−0.351636 + 0.936137i \(0.614374\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 102.280 3.99031
\(658\) 0 0
\(659\) 41.2093i 1.60528i 0.596461 + 0.802642i \(0.296573\pi\)
−0.596461 + 0.802642i \(0.703427\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) − 147.054i − 5.71111i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 66.3643 2.56579
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) − 75.3480i − 2.90015i
\(676\) 0 0
\(677\) 18.8409 0.724115 0.362058 0.932156i \(-0.382074\pi\)
0.362058 + 0.932156i \(0.382074\pi\)
\(678\) 0 0
\(679\) − 33.5611i − 1.28796i
\(680\) 0 0
\(681\) −69.1174 −2.64858
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) −7.26040 −0.275800
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 52.3643 1.97777 0.988887 0.148671i \(-0.0474996\pi\)
0.988887 + 0.148671i \(0.0474996\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 41.6434 1.56838
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −45.6113 −1.71297 −0.856484 0.516174i \(-0.827356\pi\)
−0.856484 + 0.516174i \(0.827356\pi\)
\(710\) 0 0
\(711\) − 61.0495i − 2.28954i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 4.54399i 0.169936i
\(716\) 0 0
\(717\) 73.6109 2.74905
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −49.6113 −1.84762
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3.11738 −0.115776
\(726\) 0 0
\(727\) 5.29150i 0.196251i 0.995174 + 0.0981255i \(0.0312847\pi\)
−0.995174 + 0.0981255i \(0.968715\pi\)
\(728\) 0 0
\(729\) −52.7409 −1.95336
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −53.5968 −1.97964 −0.989821 0.142318i \(-0.954545\pi\)
−0.989821 + 0.142318i \(0.954545\pi\)
\(734\) 0 0
\(735\) 51.0172i 1.88180i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 17.0961i − 0.628889i −0.949276 0.314445i \(-0.898182\pi\)
0.949276 0.314445i \(-0.101818\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −13.4164 −0.491539
\(746\) 0 0
\(747\) − 121.019i − 4.42785i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 37.8807i − 1.38229i −0.722718 0.691143i \(-0.757107\pi\)
0.722718 0.691143i \(-0.242893\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 13.0772i 0.475929i
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 26.1147i 0.945415i
\(764\) 0 0
\(765\) −136.235 −4.92558
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 14.5763i 0.524954i
\(772\) 0 0
\(773\) 46.9990 1.69044 0.845218 0.534421i \(-0.179470\pi\)
0.845218 + 0.534421i \(0.179470\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 4.25915 0.152404
\(782\) 0 0
\(783\) 9.39553i 0.335769i
\(784\) 0 0
\(785\) −30.0000 −1.07075
\(786\) 0 0
\(787\) − 6.55849i − 0.233785i −0.993145 0.116892i \(-0.962707\pi\)
0.993145 0.116892i \(-0.0372933\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −34.8247 −1.23355 −0.616777 0.787138i \(-0.711562\pi\)
−0.616777 + 0.787138i \(0.711562\pi\)
\(798\) 0 0
\(799\) − 45.6644i − 1.61549i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 4.82942i 0.170427i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 16.3643 0.575339 0.287670 0.957730i \(-0.407120\pi\)
0.287670 + 0.957730i \(0.407120\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) − 113.866i − 3.97882i
\(820\) 0 0
\(821\) 23.3765 0.815846 0.407923 0.913016i \(-0.366253\pi\)
0.407923 + 0.913016i \(0.366253\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 5.86627 0.204237
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 55.9433 1.93832
\(834\) 0 0
\(835\) 56.5055i 1.95545i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −28.6113 −0.986596
\(842\) 0 0
\(843\) 37.0803i 1.27711i
\(844\) 0 0
\(845\) −42.1956 −1.45157
\(846\) 0 0
\(847\) 28.7604i 0.988221i
\(848\) 0 0
\(849\) 90.3643 3.10130
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −40.2492 −1.37811 −0.689054 0.724710i \(-0.741974\pi\)
−0.689054 + 0.724710i \(0.741974\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −49.1935 −1.68042 −0.840209 0.542263i \(-0.817568\pi\)
−0.840209 + 0.542263i \(0.817568\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −33.6113 −1.14282
\(866\) 0 0
\(867\) 152.768i 5.18827i
\(868\) 0 0
\(869\) 2.88262 0.0977863
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 96.7031 3.27290
\(874\) 0 0
\(875\) − 29.5804i − 1.00000i
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 107.298i 3.61908i
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 47.6235i 1.59904i 0.600639 + 0.799521i \(0.294913\pi\)
−0.600639 + 0.799521i \(0.705087\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) − 9.44795i − 0.316518i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 26.4575i 0.884377i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 59.1608i 1.96008i 0.198789 + 0.980042i \(0.436299\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 0 0
\(913\) 5.71425 0.189114
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) − 51.7371i − 1.70665i −0.521380 0.853325i \(-0.674583\pi\)
0.521380 0.853325i \(-0.325417\pi\)
\(920\) 0 0
\(921\) −37.1174 −1.22306
\(922\) 0 0
\(923\) 66.7972i 2.19866i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 142.950i − 4.69510i
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 6.43270i − 0.210372i
\(936\) 0 0
\(937\) −49.3455 −1.61205 −0.806024 0.591883i \(-0.798385\pi\)
−0.806024 + 0.591883i \(0.798385\pi\)
\(938\) 0 0
\(939\) − 114.946i − 3.75113i
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −89.1530 −2.90015
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) −75.7409 −2.45865
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) − 45.6709i − 1.47788i
\(956\) 0 0
\(957\) −0.731495 −0.0236459
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 92.0020i 2.94642i
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −75.2470 −2.40245
\(982\) 0 0
\(983\) 52.9548i 1.68900i 0.535559 + 0.844498i \(0.320101\pi\)
−0.535559 + 0.844498i \(0.679899\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 49.2731i − 1.56838i
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) − 35.4965i − 1.12758i −0.825917 0.563791i \(-0.809342\pi\)
0.825917 0.563791i \(-0.190658\pi\)
\(992\) 0 0
\(993\) −115.696 −3.67150
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 14.1479 0.448069 0.224034 0.974581i \(-0.428077\pi\)
0.224034 + 0.974581i \(0.428077\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 560.2.e.c.559.7 yes 8
4.3 odd 2 inner 560.2.e.c.559.1 8
5.2 odd 4 2800.2.k.p.2351.7 8
5.3 odd 4 2800.2.k.p.2351.1 8
5.4 even 2 inner 560.2.e.c.559.2 yes 8
7.6 odd 2 inner 560.2.e.c.559.2 yes 8
8.3 odd 2 2240.2.e.d.2239.8 8
8.5 even 2 2240.2.e.d.2239.2 8
20.3 even 4 2800.2.k.p.2351.8 8
20.7 even 4 2800.2.k.p.2351.2 8
20.19 odd 2 inner 560.2.e.c.559.8 yes 8
28.27 even 2 inner 560.2.e.c.559.8 yes 8
35.13 even 4 2800.2.k.p.2351.7 8
35.27 even 4 2800.2.k.p.2351.1 8
35.34 odd 2 CM 560.2.e.c.559.7 yes 8
40.19 odd 2 2240.2.e.d.2239.1 8
40.29 even 2 2240.2.e.d.2239.7 8
56.13 odd 2 2240.2.e.d.2239.7 8
56.27 even 2 2240.2.e.d.2239.1 8
140.27 odd 4 2800.2.k.p.2351.8 8
140.83 odd 4 2800.2.k.p.2351.2 8
140.139 even 2 inner 560.2.e.c.559.1 8
280.69 odd 2 2240.2.e.d.2239.2 8
280.139 even 2 2240.2.e.d.2239.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
560.2.e.c.559.1 8 4.3 odd 2 inner
560.2.e.c.559.1 8 140.139 even 2 inner
560.2.e.c.559.2 yes 8 5.4 even 2 inner
560.2.e.c.559.2 yes 8 7.6 odd 2 inner
560.2.e.c.559.7 yes 8 1.1 even 1 trivial
560.2.e.c.559.7 yes 8 35.34 odd 2 CM
560.2.e.c.559.8 yes 8 20.19 odd 2 inner
560.2.e.c.559.8 yes 8 28.27 even 2 inner
2240.2.e.d.2239.1 8 40.19 odd 2
2240.2.e.d.2239.1 8 56.27 even 2
2240.2.e.d.2239.2 8 8.5 even 2
2240.2.e.d.2239.2 8 280.69 odd 2
2240.2.e.d.2239.7 8 40.29 even 2
2240.2.e.d.2239.7 8 56.13 odd 2
2240.2.e.d.2239.8 8 8.3 odd 2
2240.2.e.d.2239.8 8 280.139 even 2
2800.2.k.p.2351.1 8 5.3 odd 4
2800.2.k.p.2351.1 8 35.27 even 4
2800.2.k.p.2351.2 8 20.7 even 4
2800.2.k.p.2351.2 8 140.83 odd 4
2800.2.k.p.2351.7 8 5.2 odd 4
2800.2.k.p.2351.7 8 35.13 even 4
2800.2.k.p.2351.8 8 20.3 even 4
2800.2.k.p.2351.8 8 140.27 odd 4